# Properties

 Label 5070.2.a.bz Level $5070$ Weight $2$ Character orbit 5070.a Self dual yes Analytic conductor $40.484$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$40.4841538248$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.131472.2 Defining polynomial: $$x^{4} - 2x^{3} - 19x^{2} + 20x + 52$$ x^4 - 2*x^3 - 19*x^2 + 20*x + 52 Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - \beta_1 q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 + q^5 - q^6 - b1 * q^7 - q^8 + q^9 $$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - \beta_1 q^{7} - q^{8} + q^{9} - q^{10} + (\beta_{3} + 1) q^{11} + q^{12} + \beta_1 q^{14} + q^{15} + q^{16} + 4 q^{17} - q^{18} + ( - \beta_{3} - \beta_1) q^{19} + q^{20} - \beta_1 q^{21} + ( - \beta_{3} - 1) q^{22} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{23} - q^{24} + q^{25} + q^{27} - \beta_1 q^{28} + ( - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{29} - q^{30} + (\beta_{3} - 3 \beta_{2} - \beta_1) q^{31} - q^{32} + (\beta_{3} + 1) q^{33} - 4 q^{34} - \beta_1 q^{35} + q^{36} + ( - \beta_{3} + 3 \beta_{2} + 2) q^{37} + (\beta_{3} + \beta_1) q^{38} - q^{40} + (2 \beta_1 - 2) q^{41} + \beta_1 q^{42} + ( - 4 \beta_{2} + \beta_1 + 3) q^{43} + (\beta_{3} + 1) q^{44} + q^{45} + ( - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{46} + (2 \beta_{2} + 2 \beta_1 - 3) q^{47} + q^{48} + (4 \beta_{2} + \beta_1 + 3) q^{49} - q^{50} + 4 q^{51} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{53} - q^{54} + (\beta_{3} + 1) q^{55} + \beta_1 q^{56} + ( - \beta_{3} - \beta_1) q^{57} + (\beta_{3} - \beta_{2} + \beta_1 - 2) q^{58} + ( - 2 \beta_{2} + \beta_1 + 1) q^{59} + q^{60} + (2 \beta_{2} + 4) q^{61} + ( - \beta_{3} + 3 \beta_{2} + \beta_1) q^{62} - \beta_1 q^{63} + q^{64} + ( - \beta_{3} - 1) q^{66} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{67} + 4 q^{68} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{69} + \beta_1 q^{70} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{71} - q^{72} + (4 \beta_{2} - 2 \beta_1 - 2) q^{73} + (\beta_{3} - 3 \beta_{2} - 2) q^{74} + q^{75} + ( - \beta_{3} - \beta_1) q^{76} + ( - \beta_{3} - 6 \beta_{2} - \beta_1 - 2) q^{77} + ( - 3 \beta_1 - 1) q^{79} + q^{80} + q^{81} + ( - 2 \beta_1 + 2) q^{82} + ( - 2 \beta_{3} + 2 \beta_1 - 6) q^{83} - \beta_1 q^{84} + 4 q^{85} + (4 \beta_{2} - \beta_1 - 3) q^{86} + ( - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{87} + ( - \beta_{3} - 1) q^{88} + ( - \beta_{3} + 4 \beta_{2} + \beta_1) q^{89} - q^{90} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{92} + (\beta_{3} - 3 \beta_{2} - \beta_1) q^{93} + ( - 2 \beta_{2} - 2 \beta_1 + 3) q^{94} + ( - \beta_{3} - \beta_1) q^{95} - q^{96} + ( - 2 \beta_{2} + 2 \beta_1 + 8) q^{97} + ( - 4 \beta_{2} - \beta_1 - 3) q^{98} + (\beta_{3} + 1) q^{99}+O(q^{100})$$ q - q^2 + q^3 + q^4 + q^5 - q^6 - b1 * q^7 - q^8 + q^9 - q^10 + (b3 + 1) * q^11 + q^12 + b1 * q^14 + q^15 + q^16 + 4 * q^17 - q^18 + (-b3 - b1) * q^19 + q^20 - b1 * q^21 + (-b3 - 1) * q^22 + (b3 - b2 - b1 + 2) * q^23 - q^24 + q^25 + q^27 - b1 * q^28 + (-b3 + b2 - b1 + 2) * q^29 - q^30 + (b3 - 3*b2 - b1) * q^31 - q^32 + (b3 + 1) * q^33 - 4 * q^34 - b1 * q^35 + q^36 + (-b3 + 3*b2 + 2) * q^37 + (b3 + b1) * q^38 - q^40 + (2*b1 - 2) * q^41 + b1 * q^42 + (-4*b2 + b1 + 3) * q^43 + (b3 + 1) * q^44 + q^45 + (-b3 + b2 + b1 - 2) * q^46 + (2*b2 + 2*b1 - 3) * q^47 + q^48 + (4*b2 + b1 + 3) * q^49 - q^50 + 4 * q^51 + (b3 + 2*b2 + b1 + 2) * q^53 - q^54 + (b3 + 1) * q^55 + b1 * q^56 + (-b3 - b1) * q^57 + (b3 - b2 + b1 - 2) * q^58 + (-2*b2 + b1 + 1) * q^59 + q^60 + (2*b2 + 4) * q^61 + (-b3 + 3*b2 + b1) * q^62 - b1 * q^63 + q^64 + (-b3 - 1) * q^66 + (-2*b2 + 2*b1 - 4) * q^67 + 4 * q^68 + (b3 - b2 - b1 + 2) * q^69 + b1 * q^70 + (2*b3 - 2*b2 - 2*b1 - 2) * q^71 - q^72 + (4*b2 - 2*b1 - 2) * q^73 + (b3 - 3*b2 - 2) * q^74 + q^75 + (-b3 - b1) * q^76 + (-b3 - 6*b2 - b1 - 2) * q^77 + (-3*b1 - 1) * q^79 + q^80 + q^81 + (-2*b1 + 2) * q^82 + (-2*b3 + 2*b1 - 6) * q^83 - b1 * q^84 + 4 * q^85 + (4*b2 - b1 - 3) * q^86 + (-b3 + b2 - b1 + 2) * q^87 + (-b3 - 1) * q^88 + (-b3 + 4*b2 + b1) * q^89 - q^90 + (b3 - b2 - b1 + 2) * q^92 + (b3 - 3*b2 - b1) * q^93 + (-2*b2 - 2*b1 + 3) * q^94 + (-b3 - b1) * q^95 - q^96 + (-2*b2 + 2*b1 + 8) * q^97 + (-4*b2 - b1 - 3) * q^98 + (b3 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 + 4 * q^5 - 4 * q^6 - 2 * q^7 - 4 * q^8 + 4 * q^9 $$4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9} - 4 q^{10} + 2 q^{11} + 4 q^{12} + 2 q^{14} + 4 q^{15} + 4 q^{16} + 16 q^{17} - 4 q^{18} + 4 q^{20} - 2 q^{21} - 2 q^{22} + 4 q^{23} - 4 q^{24} + 4 q^{25} + 4 q^{27} - 2 q^{28} + 8 q^{29} - 4 q^{30} - 4 q^{31} - 4 q^{32} + 2 q^{33} - 16 q^{34} - 2 q^{35} + 4 q^{36} + 10 q^{37} - 4 q^{40} - 4 q^{41} + 2 q^{42} + 14 q^{43} + 2 q^{44} + 4 q^{45} - 4 q^{46} - 8 q^{47} + 4 q^{48} + 14 q^{49} - 4 q^{50} + 16 q^{51} + 8 q^{53} - 4 q^{54} + 2 q^{55} + 2 q^{56} - 8 q^{58} + 6 q^{59} + 4 q^{60} + 16 q^{61} + 4 q^{62} - 2 q^{63} + 4 q^{64} - 2 q^{66} - 12 q^{67} + 16 q^{68} + 4 q^{69} + 2 q^{70} - 16 q^{71} - 4 q^{72} - 12 q^{73} - 10 q^{74} + 4 q^{75} - 8 q^{77} - 10 q^{79} + 4 q^{80} + 4 q^{81} + 4 q^{82} - 16 q^{83} - 2 q^{84} + 16 q^{85} - 14 q^{86} + 8 q^{87} - 2 q^{88} + 4 q^{89} - 4 q^{90} + 4 q^{92} - 4 q^{93} + 8 q^{94} - 4 q^{96} + 36 q^{97} - 14 q^{98} + 2 q^{99}+O(q^{100})$$ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 + 4 * q^5 - 4 * q^6 - 2 * q^7 - 4 * q^8 + 4 * q^9 - 4 * q^10 + 2 * q^11 + 4 * q^12 + 2 * q^14 + 4 * q^15 + 4 * q^16 + 16 * q^17 - 4 * q^18 + 4 * q^20 - 2 * q^21 - 2 * q^22 + 4 * q^23 - 4 * q^24 + 4 * q^25 + 4 * q^27 - 2 * q^28 + 8 * q^29 - 4 * q^30 - 4 * q^31 - 4 * q^32 + 2 * q^33 - 16 * q^34 - 2 * q^35 + 4 * q^36 + 10 * q^37 - 4 * q^40 - 4 * q^41 + 2 * q^42 + 14 * q^43 + 2 * q^44 + 4 * q^45 - 4 * q^46 - 8 * q^47 + 4 * q^48 + 14 * q^49 - 4 * q^50 + 16 * q^51 + 8 * q^53 - 4 * q^54 + 2 * q^55 + 2 * q^56 - 8 * q^58 + 6 * q^59 + 4 * q^60 + 16 * q^61 + 4 * q^62 - 2 * q^63 + 4 * q^64 - 2 * q^66 - 12 * q^67 + 16 * q^68 + 4 * q^69 + 2 * q^70 - 16 * q^71 - 4 * q^72 - 12 * q^73 - 10 * q^74 + 4 * q^75 - 8 * q^77 - 10 * q^79 + 4 * q^80 + 4 * q^81 + 4 * q^82 - 16 * q^83 - 2 * q^84 + 16 * q^85 - 14 * q^86 + 8 * q^87 - 2 * q^88 + 4 * q^89 - 4 * q^90 + 4 * q^92 - 4 * q^93 + 8 * q^94 - 4 * q^96 + 36 * q^97 - 14 * q^98 + 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 19x^{2} + 20x + 52$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} - \nu - 10 ) / 4$$ (v^2 - v - 10) / 4 $$\beta_{3}$$ $$=$$ $$( \nu^{3} - \nu^{2} - 14\nu ) / 4$$ (v^3 - v^2 - 14*v) / 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$4\beta_{2} + \beta _1 + 10$$ 4*b2 + b1 + 10 $$\nu^{3}$$ $$=$$ $$4\beta_{3} + 4\beta_{2} + 15\beta _1 + 10$$ 4*b3 + 4*b2 + 15*b1 + 10

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 4.64466 2.32258 −1.32258 −3.64466
−1.00000 1.00000 1.00000 1.00000 −1.00000 −4.64466 −1.00000 1.00000 −1.00000
1.2 −1.00000 1.00000 1.00000 1.00000 −1.00000 −2.32258 −1.00000 1.00000 −1.00000
1.3 −1.00000 1.00000 1.00000 1.00000 −1.00000 1.32258 −1.00000 1.00000 −1.00000
1.4 −1.00000 1.00000 1.00000 1.00000 −1.00000 3.64466 −1.00000 1.00000 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5070.2.a.bz 4
13.b even 2 1 5070.2.a.ca 4
13.d odd 4 2 5070.2.b.ba 8
13.f odd 12 2 390.2.bb.c 8
39.k even 12 2 1170.2.bs.f 8
65.o even 12 2 1950.2.y.k 8
65.s odd 12 2 1950.2.bc.g 8
65.t even 12 2 1950.2.y.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.c 8 13.f odd 12 2
1170.2.bs.f 8 39.k even 12 2
1950.2.y.j 8 65.t even 12 2
1950.2.y.k 8 65.o even 12 2
1950.2.bc.g 8 65.s odd 12 2
5070.2.a.bz 4 1.a even 1 1 trivial
5070.2.a.ca 4 13.b even 2 1
5070.2.b.ba 8 13.d odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5070))$$:

 $$T_{7}^{4} + 2T_{7}^{3} - 19T_{7}^{2} - 20T_{7} + 52$$ T7^4 + 2*T7^3 - 19*T7^2 - 20*T7 + 52 $$T_{11}^{4} - 2T_{11}^{3} - 34T_{11}^{2} + 62T_{11} + 181$$ T11^4 - 2*T11^3 - 34*T11^2 + 62*T11 + 181 $$T_{17} - 4$$ T17 - 4 $$T_{31}^{4} + 4T_{31}^{3} - 79T_{31}^{2} - 556T_{31} - 956$$ T31^4 + 4*T31^3 - 79*T31^2 - 556*T31 - 956

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{4}$$
$3$ $$(T - 1)^{4}$$
$5$ $$(T - 1)^{4}$$
$7$ $$T^{4} + 2 T^{3} - 19 T^{2} - 20 T + 52$$
$11$ $$T^{4} - 2 T^{3} - 34 T^{2} + 62 T + 181$$
$13$ $$T^{4}$$
$17$ $$(T - 4)^{4}$$
$19$ $$T^{4} - 63 T^{2} + 72 T + 468$$
$23$ $$T^{4} - 4 T^{3} - 43 T^{2} + 16 T + 52$$
$29$ $$T^{4} - 8 T^{3} - 39 T^{2} + 148 T + 376$$
$31$ $$T^{4} + 4 T^{3} - 79 T^{2} - 556 T - 956$$
$37$ $$T^{4} - 10 T^{3} - 34 T^{2} + \cdots - 803$$
$41$ $$T^{4} + 4 T^{3} - 76 T^{2} - 160 T + 832$$
$43$ $$T^{4} - 14 T^{3} - 43 T^{2} + \cdots - 572$$
$47$ $$T^{4} + 8 T^{3} - 82 T^{2} + \cdots - 1103$$
$53$ $$T^{4} - 8 T^{3} - 75 T^{2} + 4 T + 52$$
$59$ $$T^{4} - 6 T^{3} - 31 T^{2} + 216 T - 284$$
$61$ $$(T^{2} - 8 T + 4)^{2}$$
$67$ $$T^{4} + 12 T^{3} - 52 T^{2} + \cdots + 352$$
$71$ $$T^{4} + 16 T^{3} - 100 T^{2} + \cdots - 5024$$
$73$ $$T^{4} + 12 T^{3} - 124 T^{2} + \cdots - 4544$$
$79$ $$T^{4} + 10 T^{3} - 147 T^{2} + \cdots + 3508$$
$83$ $$T^{4} + 16 T^{3} - 100 T^{2} + \cdots - 5024$$
$89$ $$T^{4} - 4 T^{3} - 115 T^{2} + \cdots - 1004$$
$97$ $$T^{4} - 36 T^{3} + 380 T^{2} + \cdots - 5408$$